Seismic Wave Damping System

ABSTRACT

A seismic wave damping system, and a corresponding method, includes elements, embedded within a host medium, the elements defining a seismic damping structure, and the elements being arranged to form a border of a protection zone. The seismic damping structure is configured to attenuate power of a seismic wave, traveling from a distal medium to the host medium, that is incident at the protection zone. The seismic damping structure is characterized by a resonance frequency. The system further includes an anti-resonance damping structure positioned within the protection zone and configured to dampen a residual wave propagating within the protection zone at the resonance frequency. Embodiment systems offer synergistic advantages because resonance frequencies of seismic wave damping structures may be predicted by calculation and an anti-resonance damping structure may be built to attenuate waves of primarily only specific resonance frequencies supported by the seismic wave damping structure.

RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No.62/769,517, filed on Nov. 19, 2018. The entire teachings of the aboveapplication are incorporated herein by reference.

GOVERNMENT SUPPORT

This invention was made with Government support under Contract No.FA8721-05-C-0002 awarded by the U.S. Air Force. The Government hascertain rights in the invention.

BACKGROUND

Each year, on average, a major magnitude-8 earthquake strikes somewherein the world. In addition, 10,000 earthquake-related deaths occurannually, where collapsing buildings claim the most lives, by far.Moreover, industry activity, such as oil extraction and wastewaterreinjection, are suspected to cause earthquake swarms that threatenhigh-value oil pipeline networks, U.S. oil storage reserves, andcivilian homes. Earthquake engineering building structural designs andmaterials have evolved over many years to attempt to minimize thedestructive effects of seismic surface waves. However, even under thebest engineering practices, significant damage and numbers of fatalitiescan still occur.

In particular, damage caused by earthquakes to critical structures, suchas nuclear power plants, regional hospitals, military installations,airport runways, pipelines, dams, and other infrastructure facilities,exacerbates an earthquake disaster and adds tremendous cost and time ofrecovery. Even low-energy earthquakes resulting from human activity cancause significant damage. For example, wastewater reinjection practicesused by the oil industry resulted in over 900 earthquakes in 2014-2015in the state of Oklahoma, with a recent 2016 earthquake of magnitude5.8. These continual earthquakes, although many may be small, canthreaten extremely high-value above- and below-ground pipelines thatcontrol oil supply, storage, and transport in the U.S. This can presentmajor economic and environmental concerns.

Certain structures have been proposed to protect buildings or otherareas from the effects of seismic waves. In one way or another, however,all of those structures previously proposed are inadequate, in that theycan only protect against a certain amount of seismic energy and stillallow passage, or support propagation, of a certain amount of seismicenergy. Previously proposed structures, therefore, are inadequate inthat they allow certain types or amounts of seismic waves to enter anarea that is intended to be protected.

SUMMARY

Earthquake engineering building practices apply primarily to newconstruction to decouple seismic energy traveling in the ground betweenthe ground and building foundation, whereas existing high-valuestructures are typically unlikely to be retrofitted because this is costprohibitive and because there is typically difficulty in accessing thestructures. Historically, there were no free standing subsurfacestructures used to protect existing high value assets from incominghazardous earthquake waves. More recently, some research groups haveinvestigated investigating the possibility of using boreholes in frontof an area to protect from seismic waves. These efforts have beenprimarily computer simulations, with one group conducting a small scaledfield test. This field test involved using a surface seismic source,which is not representative of a location for an earthquake, becauseearthquake sources are at depth. This test examined the ability of a fewholes to block the seismic energy in its near field. These attempts havebeen very limited in their usefulness and are not representative ofearthquakes, their geometries, raypaths, hypocenters, or seismicwavelengths or amplitudes.

Structures described herein can overcome these challenges by providingbroadband redirection and attenuation of ground motion amplitudes causedby earthquakes. Structures can provide for this by implementing anengineered, subsurface, seismic barrier (elastic wave attenuationstructure), for example. In some structures, a form of a metamaterial iscreated. A metamaterial is a material engineered to have a property thatis not found in nature. Metamaterials are made from assemblies ofmultiple elements fashioned from materials not found in the media inwhich they are embedded. In the case of the earth, boreholes andtrenches would be considered metamaterials since they are air-filled orspecialized, viscous, attenuating, fluid-filled.

As disclosed herein, a seismic barrier (or metamaterial) can includeborehole array complexes or trench complexes that reflect, refract,absorb, divert, or otherwise impede destructive seismic surface wavesfrom a designated “protection zone.” Seismic surface waves against whichdisclosed wave damping structures can protect include Rayleigh waves(ground-roll), shear waves, Love waves, and compressional waves.

Further, disclosed wave damping structures can overcome the limitationsof using a vertical borehole structure only in front of an intendedprotection zone. Use of a vertical borehole structure only in front ofan intended protection zone (between a seismic wave incoming toward theprotection zone and the protection zone itself) is not effective enough,since most seismic waves will diffract around a vertical boreholestructure vertically and still strike the protection area withconsiderable force. However, with respect to structures describedherein, boreholes or trenches (example “elements” as used herein) formedat an angle with respect to the vertical, with lateral offset into theearth and toward a zone and surface structure to be protected, canbetter divert seismic waves farther away from the intended protectionzone than straight, deep boreholes. In addition, seismic wave dampingstructures described herein and incorporating such borehole or trenchelements have been demonstrated, through numerical modeling and benchscale measurements, to provide broadband seismic wave amplitudereduction.

By using angles for holes that point down below a structure to beprotected, seismic wave power can be effectively diverted far underneaththe structure. Angled holes forming groupings or tapered structures canbe particularly helpful due to the vertical depths that surface wavescan reach, which can be hundreds of meters or greater. Moreover, byusing angled holes on multiple sides of a structure, an aperture betweenprotective holes can be made small, effectively blocking most seismicenergy from diffraction toward the protected zone, thereby significantlylimiting any need for deep boring, which can be cost-prohibitive. Astructural arrangement formed of at least two angled borehole elementswith opposing orientations with respect to a vertical, thus forming atapered aperture, can be referred to herein as a “muffler.” Suchstructures are described hereinafter in greater detail with respect tothe drawings.

Furthermore, retrofitting a building area with embodiment waveattenuation structures can be done with much flexibility, becausedisclosed structures can be implemented farther from a structure to beprotected, at least at the Earth's surface. Further, certain periodicgroupings of boreholes, such as sawtooth-shaped groupings or othergeometric groupings, may be employed and can increase a range of seismicwavelengths against which a disclosed structure can be made effective.Such layout geometries can advantageously be configured to causereflected self-interference of a traveling seismic wave, thus reducingthe waves's effective ground motion amplitude. Still further, structuresdescribed herein can be used in protecting areas of the ocean or oceanfront from the destructive effects of sea waves, such as tsunamis.

In one structure described herein, a seismic wave damping structureincludes a structural arrangement of at least two elements, each elementdefining an inner volume and containing therein a medium resistant topassage of an anticipated seismic wave having a wavelength at least oneorder of magnitude greater than a cross-sectional dimension of the innervolume of the elements. The structural arrangement can taper from anupper aperture to a lower aperture, the structural arrangement defininga protection zone at the upper aperture, the upper aperture being largerthan the lower aperture. The structural arrangement can be configured toattenuate power from the anticipated seismic wave within the protectionzone relative to power from the anticipated seismic wave external to theprotection zone.

The structural arrangement can be further configured to attenuate powerfrom a Rayleigh wave, or from at least one of a compressional, shear, orLove elastic wave.

The at least two elements can be boreholes in earth, and the upperaperture can be closer to a surface of the earth than the loweraperture. As an alternative, the at least two elements can be trenchesin earth, while the upper aperture can still be closer to the surface ofthe earth than the lower aperture.

The anticipated seismic wave can be a seismic wave in earth, and themedium resistant to passage of the anticipated seismic wave can be airor at least one of a gas, water, or viscous fluid. The anticipatedseismic wave can be a water wave, and the medium resistant to passage ofthe water wave can include a solid material. The upper aperture can becloser to an upper surface of the water in the absence of theanticipated water wave. As an alternative, the upper aperture can be inair, and the lower aperture can be in earth or water in absence of theanticipated water wave.

A depth of the lower aperture in earth or water can be on the order of100 meters. Each element can further include a structural lining betweenthe inner volume and an exterior of the element. A width of the upperaperture can be on the order of 0.5 km. Particular preferred dimensionsfor particular upper apertures can be predicted using expressions givenhereinafter.

Each of the at least two elements can include a plurality of discretesub-elements, each of the sub-elements defining a respective sub-elementinner volume and containing therein the medium resistant to passage ofthe seismic wave. Cross-sections of respective discrete sub-elementscorresponding to at least one of the elements can be located at pointscollectively defining a hexagon.

The damping structure can also include a plurality of structuralarrangements defining a superstructure, and the protection zone canencompass, at least partially, upper apertures of respectivearrangements of the plurality of structural arrangements. Thesuperstructure can be configured to attenuate power from the elasticwave within the protection zone relative to power from the seismic waveexternal to the protection zone. The protection zone can extend, inlength, from one of the at least two elements to the other at the upperaperture. The protection zone can have a width, measured perpendicularto the length, of approximately 5%, 10%, 25%, 50%, 75%, or 100% of thelength. The protection zone can be defined by a region, bounded at leastpartially by the at least two elements, within which the structuralarrangement is configured to attenuate power from the seismic wave by atleast 10 dB in power within the protection zone relative to power fromthe seismic wave external to the protection zone. Larger reductions inpower have also been demonstrated by the authors using numericalsimulations and scaled measurements.

The damping structure can also include an incident grouping of elementssituated at a border of the protection zone expected to receive theseismic wave, as well as a transmission grouping of elements situated ata border of the protection zone opposite the incident grouping. Thestructural arrangement of at least two elements can include one elementof the incident grouping and one element of the transmission grouping.Each element of the incident and transmission groupings of elements canhave upper and lower ends thereof, and each element of the incident andtransmission groupings of elements can define an inner volume andcontain therein the medium resistant to passage of the anticipatedseismic wave. The incident and transmission groupings can form asuperstructure.

Upper ends or lower ends of respective elements of the incident ortransmission grouping may be situated along an element row. Upper endsor lower ends of respective elements of the incident or transmissiongrouping may further be situated along a plurality of substantiallyparallel rows to form an element array. Upper ends or lower ends ofrespective elements of the incident or transmission grouping may besituated to form a substantially periodic pattern. The substantiallyperiodic pattern may be a substantially sawtooth pattern, or the patternmay be configured to cause constructive or destructive interference ofdiffracted portions of the anticipated seismic wave diffracted fromrespective elements.

The damping structure can also include an electro-mechanical generatorconfigured to generate or store electrical power using mechanical powerfrom the anticipated seismic wave.

In another disclosed seismic wave damping structure the structure mayinclude a structural grouping of elements, each element of thestructural grouping defining an inner volume and containing therein amedium resistant to passage of an anticipated elastic wave having awavelength at least one order of magnitude greater than across-sectional dimension of the inner volume of the elements. Eachelement of the structural grouping may have an upper end and a lower endthereof defining a first line from the upper end to the lower end. Thefirst line can form an acute angle with a second line defining adirection of travel of the anticipated elastic wave toward a protectionzone. The structural grouping of elements may be configured to attenuatepower from the seismic wave within the protection zone relative to powerfrom the seismic wave external to the protection zone.

The grouping of elements can be further configured to attenuate powerfrom a Rayleigh elastic wave, or from at least one of a compression,shear, or Love seismic wave. Each element may be a borehole in earth ora trench in earth, with the upper end closer to a surface of the earththan the lower end.

At least one of the elements can include a plurality of discretesub-elements substantially parallel to each other, each of thesub-elements defining a respective sub-element inner volume andcontaining therein the medium resistant to passage of the elastic wave.Cross-sections of respective discrete sub-elements may be located atpoints in a cross-sectional plane collectively defining a hexagon.

Upper ends or lower ends of respective elements can be situated along anelement row. Furthermore, upper ends or lower ends of respectiveelements can be situated along a plurality of substantially parallelrows to form an element array. Upper ends or lower ends of respectiveelements can be situated to form a substantially periodic pattern, andthe substantially periodic pattern may be a substantially sawtoothpattern. The substantially periodic pattern may also be configured tocause constructive or destructive interference of diffracted portions ofthe anticipated seismic wave diffracted from respective elements.

The grouping of elements can be an incident grouping of elementssituated at a border of the protection zone at which the anticipatedseismic wave is expected to be incident. The damping structure canfurther include a transmission grouping of elements situated at anopposite border of the protection zone opposite the incident grouping.Each element of the transmission grouping can define an inner volume andcontain therein a medium resistant to passage of the anticipated seismicwave having a wavelength at least one order of magnitude greater than across-sectional dimension of the inner volume of the element. Eachelement of the transmission grouping may have an upper end and a lowerend thereof defining a first line from the upper end to the lower end,the first line forming an obtuse angle with a second line defining adirection of travel of an attenuated anticipated seismic wave away fromthe protection zone. The transmission grouping of elements can beconfigured to attenuate power from the elastic wave within theprotection zone relative to power from the elastic (e.g., seismic) waveexternal to the protection zone and transmitted through or around theincident grouping of elements.

A separation of upper ends of elements of the incident grouping fromupper ends of respective elements of the transmission grouping can be onthe order of 0.5 km. The protection zone can be further defined by aregion, bounded at least partially by the incident and transmissiongroupings, within which the incident and transmission groupings areconfigured to attenuate power from the seismic wave by at least 10 dBwithin the protection zone relative to power from the seismic waveexternal to the protection zone.

Also disclosed is an elastic wave damping structure that can includefirst means for damping an anticipated seismic wave and second means fordamping an anticipated seismic wave, wherein a combination of the firstmeans and the second means forms an upper aperture and a lower aperture.The upper aperture can taper to lower aperture, the combination defininga protection zone at the upper aperture. The structural arrangement canbe configured to attenuate power from the anticipated seismic wavewithin the protection zone relative to power from the anticipatedseismic wave external to the protection zone.

Still another disclosed seismic wave damping structure can include firstmeans configured to attenuate power from an anticipated seismic wave andat least one second means configured to attenuate power from the seismicwave. Each of the first and second means can define a first line from anupper end of the means to a lower end of the means, the first lineforming an acute angle with a second line defining a direction of travelof the anticipated seismic wave toward a protection zone.

One limitation of the seismic wave damping structures described above isthat they do not block or attenuate all power from an anticipatedseismic wave from entering a protection zone. Some seismic power maystill enter the protection zone, particularly higher frequencies.Specifically, a seismic wave damping structure is described in theexamples above and in other parts of this specification can be veryeffective at attenuating power from an anticipated seismic wave in lowerfrequency ranges. Nonetheless, the seismic wave damping structuresdescribed above may permit propagation of frequencies, particularlyfrequencies in a higher frequency range, within the protection zone.

Advantageously, as described herein, the seismic wave damping structuresdescribed above may have resonant frequencies that can be predictedbased on parameters of the structure and based on properties of theprotection zone, particularly earth, soil, ground, etc. within theprotection zone, in which the seismic wave damping structure isembedded. As further described hereinafter, the disclosed seismic wavedamping structures may be advantageously combined with anti-resonancedamping structures described hereinafter according to variousembodiments. These combinations may form a seismic wave damping systemthat is extremely effective at preventing seismic waves from damagingprotected structures in a protection zone. While some of the dampingstructures described herein have previously been known, they have notbeen used in the sense or combination described in this specification,namely as anti-resonance damping structures. As described hereinafter,when used as anti-resonance damping structures, they may be configuredto address, specifically, resonance frequencies supported by thedisclosed seismic damping structures, as described above.

When used in combined systems as noted above, and as described furtherhereinafter, seismic wave damping structures and anti-resonance dampingstructures have particular synergistic effects when used in combinationwith each other. On one hand, anti-resonance damping structuresdescribed herein may increase effectiveness of seismic wave dampingstructures described herein in damping residual seismic waves thatpropagate within the protection zone, which may be allowed to pass theseismic wave damping structures. On the other hand, synergistically,based on properties of a host medium in which the seismic dampingstructure is embedded, resonance frequencies supported by a givenseismic wave damping structure may be specifically predicted based onone or more properties of the host medium and on one or more physicalproperties of the seismic damping structure. A correspondinganti-resonance damping structure may, therefore, be configured to dampenspecifically a residual wave propagating within the protection zone atthe resonance frequency. A synergy of this arrangement is that theanti-resonance damping structure need not be configured to address allpossible seismic frequencies, potentially requiring more complex andextensive engineering. Instead, the anti-resonance damping structure maybe built and configured to dampen, specifically, only one or moreresonance frequencies supported by the seismic damping structure, wherethese resonance frequencies may be specifically predicted based onproperties of the host medium and seismic damping structure.Accordingly, according to embodiments described hereafter, particularsynergies may be obtained, which have not been contemplated or describedbefore.

In one particular embodiment disclosed herein, a seismic wave dampingsystem includes elements, embedded within a host medium. The elementsdefine a seismic damping structure and are arranged to form a border ofa protection zone. The seismic wave damping structure (also referred toherein as “seismic damping structure”) is configured to attenuate powerof a seismic wave, traveling from a distal medium outside of theprotection zone to the host medium in which the seismic dampingstructure is embedded. Thus, the seismic wave is incident at the seismicdamping structure, at the protection zone. The seismic wave dampingstructure is characterized by one or more resonance frequencies that maybe predetermined (i.e., known by prediction or modeling or calculation).

The resonance frequency may be a function of a depth of the elementsembedded in the host medium and of a property (i.e., a physicalproperty) of the host medium.

The anti-resonance damping structure may be configured to dampen theresidual wave by being mechanically tuned to the resonance frequency. Inother words, the anti-resonance damping structure may be built accordingto dimensions and specifications that will allow it to specificallydampen the resonance frequency, or the anti-resonance damping structuremay be built, and then calibrated in such a manner that it absorbs orscatters preferentially at the resonance frequency, a harmonic of theresonance frequency, or a subharmonic of the resonance frequency. Theanti-resonance damping structure may be configured to dampen theresidual wave by being mechanically tuned to a harmonic of the resonancefrequency or to a subharmonic of the resonance frequency. Theanti-resonance damping structure may include two or more anti-resonancedamping structures that are configured to dampen the residual wave bybeing mechanically tuned to two or more respective frequencies selectedfrom the group consisting of (i) the resonance frequency, (ii) harmonicsof the resonance frequency, and (iii) sub harmonics of the resonancefrequency.

The anti-resonance damping structure may include one or more Helmholtzresonators positioned on or within the host medium within the protectionzone. The one or more Helmholtz resonators may be filled with a gas,such as air, or with water or a viscous fluid. The anti-resonancedamping structure may be an array of cylinders or other shaped elements,such as meta-concrete cylinders, buried within the host medium withinthe protection zone.

The anti-resonance damping structure may be a seismic wave absorbingstructure configured to dampen the residual wave by absorption. Theseismic wave absorbing structure may be a mass-in-mass lattice. The hostmedium may be earth, and the anti-resonance damping structure mayinclude an array of trees that are not placed in order naturally, butare rather planted in the earth in a specific configuration thatincludes being spaced periodically within the protection zone. Theanti-resonance damping structure may include an array of scatteringcomponents that are positioned periodically on or within the host mediumwithin the protection zone. The anti-resonance damping structure mayinclude one or more towers positioned on the host medium within theprotection zone. The one or more towers may be one or more flexible,steel-girded towers. The one or more towers may have heights, extendingvertically from a surface of the host medium, such as a ground surface,between a few meters and hundreds of meters. In various examples,heights may be on the order of 300 m, on the order of 200 m or, on theorder of 100 m, on the order of 50 m, on the order of 25 m, or on theorder of 10 m, for example. The one or more towers may have heights,extending vertically from a surface of the host medium, of 100 m orless, such as between about 10 m and about 100 m. Heights of the one ormore towers may be specifically configured such that the towers dampenthe residual wave at the resonance frequency by dampening one or moreharmonics or subharmonics of the resonance frequency. The one or moretowers may have, each, a cross-sectional dimension, such as a diameter,on the order of 1 m, on the order of 5 m, on the order of 10 m, or onthe order of 15 m, for example.

In another embodiment, a method of constructing a seismic waveprotection zone includes embedding elements within a host medium, thusdefining a seismic wave damping structure characterized by a resonancefrequency and forming a border of a protection zone. The seismic wavedamping structure is built or otherwise configured to attenuate power ofa seismic wave traveling from a distal medium to the host medium, wherethe seismic wave may be anticipated to be incident at the protectionzone. The method also includes positioning an anti-resonance dampingstructure within the protection zone and configuring the anti-resonancedamping structure to dampen a residual wave propagating within theprotection zone at the resonance frequency.

The method may further include positioning the anti-resonance dampingstructure within the protection zone and configuring the anti-resonancedamping structure to dampen a residual wave by building or tuning theanti-resonance damping structure to dampen the resonance frequency,where the resonance frequency is predicted based on one or moreproperties of the host medium and one or more dimensions or otherproperties of the elements embedded within the host medium forming theseismic wave damping structure. Configuring the anti-resonance dampingstructure to dampen the residual wave propagating within the protectionzone at the resonance frequency may include configuring theanti-resonance damping structure based on one or more properties of thehost medium, such as a wave propagation velocity, and one or moreproperties of the elements, such as length (depth) in the host medium,such as depth in earth.

In a further embodiment, a method of seismic wave damping includesconverting an incident seismic wave propagating in a distal mediumoutside a protection zone into a residual seismic wave propagatingwithin the protection zone at one or more resonant frequencies. Themethod further includes dampening the residual wave within theprotection zone via anti-resonance damping. The method may optionallyinclude use or incorporation of any of the methods; elements; seismicwave damping structures, superstructures, or arrangements; andanti-resonance damping structures summarized hereinabove pertaining toother embodiments or further described hereinafter in relation to otherembodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawings will be provided by the Office upon request and paymentof the necessary fee.

The foregoing will be apparent from the following more particulardescription of example embodiments of the invention, as illustrated inthe accompanying drawings in which like reference characters refer tothe same parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingembodiments of the present invention.

FIG. 1A is a cross-sectional side view of an embodiment elastic wavedamping structure that includes a downward-tapered, V-shaped structuralarrangement with two elements.

FIG. 1B is a diagram of the embodiment of FIG. 1A without theabove-ground building structure and with additional below-ground detailsas compared to FIG. 1A.

FIG. 2A is a top-view illustration of the structural arrangement of FIG.1B.

FIG. 2B is a top-view illustration of a linear structural grouping ofelements forming an embodiment elastic wave damping structure.

FIG. 2C is a top-view illustration of an embodiment elastic wave dampingstructure including an arrangement of two elements formed of pluralitiesof discrete sub-elements in hexagonal clusters.

FIG. 2D is a top-view illustration two structural groupings of elements,namely an incident structural grouping to 304 a of elements that arearranged into substantially parallel element rows 226 on the incidentside of the protection zone.

FIG. 2E is a top-view illustration of an embodiment wave dampingstructure including a superstructure with two of the structuralarrangements illustrated in FIG. 1B.

FIG. 2F is a top-view illustration of a superstructure includes trenchelements instead of borehole elements.

FIG. 3 is a cross-sectional side-view diagram showing how embodimentscan be advantageously used to protect structures in the sea, such as anoil platform.

FIG. 4A is a cross-sectional, side-view illustration of incident andtransmission structural element groupings, respectively, of elements oneither side of a protection zone.

FIG. 4B is a cross-sectional, side-view illustration of an incidentstructural grouping of elements that can be employed to protect againstincident water waves, such as tsunami waves.

FIG. 5 is a top-view illustration of an arrangement of elements formingan incident structural grouping of the elements in a sawtooth pattern.

FIG. 6 is a top-view illustration of an embodiment wave dampingstructure with a substantially sawtooth, substantially periodic patternof elements that form an incident structural grouping.

FIG. 7 is a top-view illustration of an additional incident structuralgrouping of elements showing interference between reflected wavelets andshowing constructive interference used to generate electrical power.

FIG. 8A is a top-view illustration of a protection zone formed betweenmodeled arrays of borehole elements.

FIG. 8B is a three-dimensional (3D) illustration of the protection zoneand borehole elements shown in FIG. 8A.

FIG. 8C is a 3D view of modeled trench elements in a 3D finite elementmeshing.

FIG. 9A shows model equations used in finite element analysis ofembodiments.

FIG. 9B is a 3D representation of the finite element analysis.

FIG. 9C is a graph showing a source time function for rupture velocityof an earthquake measured and used as a source time function for thefinite element analysis.

FIG. 9D is a graph showing power reduction calculated in a protectionzone with and without embodiment wave damping structures.

FIG. 9E is collection of areal and depth view seismic wave snapshotscalculated with and without embodiment seismic wave damping structures.

FIG. 10 is a table showing various high-value assets that may beprotected, as examples, using embodiment elastic wave dampingstructures, with expected upper aperture and lateral extent sizes andnumbers of boreholes for corresponding structures.

FIG. 11 is a series of diagrams and a graph showing the comparativeeffects of seismic cloaking on seismic wave propagation for a single,vertical barrier element compared with an embodiment angled structuralelement, as calculated using a finite difference 2D model of the barrierstructures.

FIGS. 12A-12B show a diagram and equations illustrating semi-analyticalsimulation of acoustic propagation through a “tapered muffler” geometry.

FIGS. 13A and 13B are graphs showing transmission loss as a function offrequency for various acoustic “muffler” parameters, as calculated usingthe analytical tools shown in FIGS. 12A-12B.

FIG. 14 is a collage of photographs showing apparatus used to study, ona model scale, non-naturally occurring, man-made structures such asborehole arrays and trenches embedded in elastic media analogous to rockand compact soil using a machined, table-top scaled physical model.

FIGS. 15A-15C show measured results that illustrate the effects of amodel V-trench machined in Delrin® block table-top experimentalconfiguration. In particular, FIG. 15A and FIG. 15B show accelerometertime series traces measured in a center line across a homogeneous solidDelrin® block relative to the transducer source location, and FIG. 15Cshows the model trench barrier structure in schematic form, along withaccelerometer locations corresponding to the traces in FIGS. 15A-15B.

FIG. 16 shows earthquake magnitude reduction expected due to subsurfacebarrier structures, based on extrapolation from the measured andmodelled results.

FIG. 17A is a schematic diagram illustrating an embodiment seismic wavedamping system in its environment of use.

FIG. 17B is a schematic diagram illustrating, more particularly, theseismic wave damping system of FIG. 17A, without it's contextualenvironment of use.

FIG. 18A is a perspective, colored or shaded illustration of a V-shapedseismic muffler (seismic damping structure) used in connection with theanti-resonance damping structure of FIGS. 17A-17B to form an embodimentseismic wave damping system.

FIG. 18B is a color-coded or shaded, side-view graphical drawingillustrating particle velocity, in unit length per second, over across-sectional area showing the seismic damping structure of FIG. 18A.

FIG. 19A is a cross-sectional diagram illustrating geometry and termsterminology for an example conical shaped muffler seismic dampingstructure.

FIG. 19B is a diagram illustrating for different muffler geometries andtheir respective dimensions.

FIG. 19C is a graph illustrating elastic wave transmission loss as afunction of wavelength corresponding to seismic frequencies spanningfrom 0.1-10 Hz for the muffler geometries illustrated in FIG. 19B.

FIG. 19D is a graph illustrating P and S wave transmission loss behaviorfor the four muffler examples illustrated in FIG. 19B as a function ofseismic frequency.

FIG. 20A illustrates four different muffler geometries, wherein theinlet diameter is 0.1 km and the muffler wall slope is constant for allcases except the last case, which shows vertical a vertical wall mufflerfor comparison.

FIG. 20B is a graph illustrating wave transmission loss as a function ofseismic wavelength corresponding to seismic frequencies from 0.1 Hz to10 Hz.

FIG. 20C is a graph illustrating P and S wave transmission loss for thefour muffler examples illustrated in FIG. 20A as a function of seismicwave incident frequency.

FIG. 21A is a diagram illustrating cross-sectional muffler geometry fora shallow sloping muffler model (a) having a wall slope and for avertical wall muffler model (d).

FIG. 21B is a cross-sectional illustration of the example structures inFIG. 21A, along with color-coded or shaded illustration of numericallycalculated damping characteristics.

FIG. 21C is a graph illustrating transmission loss as a function ofseismic frequency for the two example structures illustrated in FIG.21A.

FIG. 22A illustrates a time source a source time function from theHector Mine 1999 earthquake.

FIG. 22B is a graph showing amplitude as a function of frequency for theHector Mine earthquake.

FIG. 23A is a color-coded or shaded, cross-sectional illustration of aseismic wave field in presence of an embodiment seismic wave dampingsystem that includes a Helmholtz resonator array anti-seismic dampingstructure as part of the system.

FIG. 23B is a graph showing the source function for the Hector Mineearthquake, injected by simulation into the model represented in thegraph of FIG. 23A.

FIG. 23C illustrates calculations by which damping of a Helmholtzresonator array may be determined.

FIG. 23D is a schematic diagram illustrating the Helmholtz resonatorarray, anti-resonance damping structure graphically illustrated in FIG.23A.

FIG. 23E is a graph illustrating seismic amplitude as a function offrequency, as reduced by a seismic wave damping structure alone, and asreduced by the seismic wave damping structure in combination with theHelmholtz resonator array example anti-resonance damping structure ofFIG. 23A.

FIG. 24A is a color-coded or shaded, vertical view cross-sectional graphof a seismic wave damping structure embodiment including a tower or treearray example anti-resonance damping structure in presence of a shearwave of the Hector Mine earthquake.

FIG. 24B is a graph showing the source function for the Hector Mineearthquake, which was injected into the model illustrated in FIG. 24A.

FIG. 24C shows an equation that can be used to calculate dampingfrequency for the array of resonators (towers or trees, e.g.)illustrated in FIG. 24A.

FIG. 24D is a more detailed illustration of the array of resonatorsillustrated in FIG. 24A.

FIG. 24E is a graph illustrating damping, particularly seismic amplitudedecrease, obtained by using the combination of seismic wave dampingstructure and anti-resonance damping structure of the embodiment systemillustrated in FIG. 24A.

FIG. 25A is a cross-sectional illustration of a meta-concrete arrayexample of buried cylinders used as an anti-resonance damping structureas part of an embodiment system, illustrated in presence of the HectorMine earthquake source function used in the model.

FIG. 25B is a graph illustrating the Hector Mine earthquake sourcefunction injected into the model illustrated in FIG. 25A.

FIG. 25C is a cross-sectional view of the meta-concrete array exampleanti-resonance damping structure illustrated in FIG. 25A.

FIG. 25D is an equation illustrating how resonance frequency of themedic concrete array of FIG. 25A may be calculated.

FIG. 25E is a graph illustrating seismic amplitude reduction dampingthat may be obtained using the meta-concrete array example seismicanti-resonance damping structure as part of an embodiment seismic wavedamping system illustrated in FIG. 25A.

FIG. 26 is a flow diagram illustrating an embodiment procedure forseismic wave damping.

DETAILED DESCRIPTION

A description of example embodiments of the invention follows.

Seismic Wave Damping Structures

FIG. 1A is a cross-sectional side view of an embodiment elastic wavedamping structure that includes a downward-tapered, V-shaped structuralarrangement with two elements 30 a and 30 b. The structural arrangementdefines a protection zone 40 between the elements 30 a and 30 b, where abuilding 20 that is desired to be protected is located above theprotection zone. The damping structure is configured to protect thebuilding 20 from incident surface waves 60 that initially impinge on theelement 30 a as a result of an earthquake 50. A small amount of seismicwave power still enters the protection zone 40, such that there areattenuated surface waves 60′ under the building 20. Some wave energyalso travels underneath and around the elements 30 a and 30 b to becometransmitted surface waves 60″.

FIG. 1B is a cross-sectional side view of an embodiment elastic wavedamping structure that includes a structural arrangement 100 with twoelements 102 a and 102 b. Each of the elements 102 a and 102 b definesan inner volume 116 that contains therein a medium resistant to passageof an anticipated incident elastic wave 114. The wave 114 has awavelength at least one order of magnitude greater than across-sectional dimension of the inner volume of the elements, such asthe diameter d of the element 102 a. The element 102 a may be referredto as having an upper end 108 and a lower end 110. Likewise, while notmarked in FIG. 1B, the element 102 b has upper and lower ends similar tothose of element 102 a.

The structural arrangement 100 tapers from an upper aperture 104 to alower aperture 106. The structural arrangement 100 defines a protectionzone, also referred to herein as a “structural protection zone” 112, atthe upper aperture 104. The structural arrangement 100 is configured toattenuate power from the anticipated, incident elastic wave 114 withinthe protection zone 112, relative to power from the anticipated elasticwave external to the protection zone. Thus, outside of the protectionzone 112, the incident elastic wave 114 has a given power, which can bereferred to as seismic power, where the elastic wave 114 propagates inearth 118. However, due to the attenuation of power within theprotection zone 112, which is caused by the structural arrangement 100,a component 114′ of the elastic wave 114, which is transmitted into theprotection zone 112, has an attenuated seismic power relative to theincident elastic wave 114 that is incident at the structural arrangement100. In particular, the elastic wave 114 is incident at the element 102a of the structural arrangement 100.

The upper ends of the elements 102 a and 102 b are located at greater(more positive) Z values along the Z axis that is illustrated in FIG.1B, while the lower ends 110 are located at smaller positive (morenegative), Z values than the upper ends 108.

The structural arrangement 100 can be configured to attenuate power froma Rayleigh wave, or from at least one of a compressional, share, or loveelastic wave. Thus, where the elastic weight 114 is a seismic wave, forexample, the structural arrangement 100 is, advantageously, effectivefor attenuating surface seismic waves, as well as seismic waves of othertypes.

The elements 102 a-b can be boreholes in earth, and, as describedhereinabove, the upper aperture 104 can be closer to a surface 120 ofthe earth 118 than the lower aperture 110. However, in otherembodiments, the two elements 102 a and 102 b can be trenches in theearth 118, as described hereinafter in connection with FIG. 2F. Foreither boreholes or trenches, or other configurations of the elements102 a-b, the upper aperture 104 can be closer to a surface 120 of theearth 118 than the lower aperture 110, as illustrated in FIG. 1B. Thelower aperture 110 is at a depth 119 in the earth below the surface 120.In many embodiments, the length of the upper aperture, as alsoillustrated in FIG. 2A and described below, can be on the order of 0.5km, for example.

Furthermore, continuing to refer to FIG. 1B, in cases where theanticipated incident elastic wave 114 is a seismic wave in the earth,the medium contained in the inner volume 116 and resistant to passage ofthe wave 114 can be air, for example. However, in other embodiments, themedium contained in the inner volume 116 can be at least one of a gas,water, or viscous fluid. In all of these cases of air, gas, water, orviscous fluid, for example, the medium does not transmit seismic powernearly as well as the earth 118. Hence, the medium resists passage ofthe wave 114, and seismic power can be deflected, reflected, diffracted,or otherwise dissipated before it enters the protection zone 112 as theattenuated wave 114 with attenuated seismic power. Moreover, given thetapered shape of the structural arrangement 100, which can also beconsidered to be angles of the elements 102 a and 102 b with respect tothe z-axis, seismic power can be directed downward, away from anystructures or objects desired to be protected within the structuralprotection zone 112.

The limited-size, lower aperture 106 helps to prevent leakage of seismicpower around the element 102 a, at which the seismic wave 114 isincident, and up to the protection zone. Such leakage presents somelimitation on the effectiveness of the element 102 a as a seismicshield; however, it should also be recognized that, in otherembodiments, such as that illustrated in FIGS. 4B and 5-7, a grouping ofelements, such as the element 102 a, may still be advantageously used onan incident side of the structural protection zone 112 in order to limitseismic power reaching the protection zone. Thus, while the element 102b on the side of the protection zone opposite the incident side causesthe structural arrangement 100 to be much more effective, embodimentscan nevertheless employ an array of elements 102 b on the incident sideof the protection zone, where the elastic wave 114 is expected to beincident, with some helpful attenuation and deflection of seismic powerreaching the protection zone.

The structural arrangement 100 illustrates a key feature of manyembodiments, the ability to be effective in broadband shielding againsta wide range of seismic wavelengths. In particular, broadbandwavelengths longer than the lower aperture 106 are most effectivelyblocked by the structural arrangement 100. While higher frequency,shorter wavelength seismic waves with wavelengths shorter than the loweraperture 106 are not as effectively shielded by the arrangement 100, themajority of seismic power is typically present at the very lowfrequencies and longer wavelengths that are less able to enter throughthe lower aperture.

FIG. 2A is a top-view of the structural arrangement 100 illustrated inFIG. 1B, viewing the X-Y plane, down along a line of sight following theZ-axis illustrated in FIG. 1B. As illustrated in FIG. 2A, a length 222of the upper aperture 104 can be considered to extend the entire lengthbetween the upper end of the element 102 a and the upper end of theelement 102 b. This is because, between the elements 102 a and 102 b,there is some degree of attenuation of the incident wave 114 at allpoints, even though the attenuation can vary. Although thecross-sectional profiles of the elements 102 a and 102 b are round, itshould be understand that cross-sectional profiles of elements in otherembodiments may have any shape, such as oval, rectangular, or any othershape. However, if drilling is used to form the elements, it is likelymost convenient to bore out elements with circular cross sections.

The protection zone 112 can also have a width 224 that can extend,measured perpendicular to the length 222, approximately 5%, 10%, 25%,50%, 75%, or 100% of the length 222, for example the region defined asthe protection zone 112 can also be defined in terms of a particulardegree of attenuation of the seismic waves 114 that can be achieved. Forexample, the protection zone 112 can be defined by a region, bounded byat least partially by the elements 102 a and 102 b, within which thestructural arrangement 100 is configured to attenuate power from theelastic wave 114 by at least 10 dB. This attenuation can be within theprotection zone 112, relative to power from the elastic wave 114external to the protection zone 112. Furthermore, other criteria can beused to select the protection zone, such as the region within which a 3dB attenuation of seismic power is obtained, or within which more thananother given value, such as more than 30 dB of attenuation is obtained,for example.

FIG. 2B illustrates that a structural arrangement can have more thanjust two elements, such as the elements 102 a and 102 b in FIG. 1B andFIG. 2A. In particular, four elements 102 a are oriented along a line(element row) 226, in the direction of the y-axis, between the incidentwave 114 and the protection zone 112. The four elements 102 a form astructural grouping 232 of elements. Because these elements are on aside of the production zone 112 that is expected to receive the incidentseismic wave 114, the structural grouping 232 can be referred to as an“incident” structural grouping of elements.

FIG. 2C is a top view of two elements 202 a and 202 b. The element 202 ais formed of a plurality of discrete sub-elements 208 a. The discreteset of elements 208 a can be smaller than the element 102 a in FIG. 1B,or they may be of the same size as element 102 a. The element 202 aforms a cluster with the sub-elements situated to increase structuralstrength. In particular, in FIG. 2C, the sub-elements 228 a are locatedsuch that cross-sections of the respective sub-elements 208 acorresponding to the element 202 a are located at points collectivelydefining a hexagon 230. Similarly, the element 202 b is formed ofdiscrete sub-elements 208 b located at points defining the hexagon 230.The cross sections illustrated in FIG. 2C are located in across-sectional plane that is the X-Y plane shown in FIG. 2C. However,another example cross-sectional plane in which the elements can belocated at points forming a hexagon is a plane perpendicular to theelements 202 a, which do not extend downward along the Z axis.

As is understood in the art of mechanical engineering, the hexagonformation for structural elements can be particularly strong. However,it should be understood that discrete sub-elements can be arranged inother orientations, such as in groups of four, in pentagon or otherpolygon shapes, or in other arrangements, for example.

FIGS. 2D-2F illustrate other embodiment arrangements of elements. Inparticular, the arrangements illustrated in the top-view illustrationsof FIGS. 2D-2F, in which there are a plurality of structuralarrangements, constitute other example orientation patterns.

FIG. 2D, in particular, illustrates two structural groupings ofelements, namely an incident structural grouping to 234 a of elementsthat are arranged into substantially parallel element rows 226 on theincident side of the protection zone 112, which is the side at which theanticipated seismic wave 114 is anticipated to arrive toward theprotection zone. Similarly, at the opposite side of the protection zone112, the superstructure 240 a includes two additional, substantiallyparallel element rows 226 of elements 102 b that include a transmissionstructural grouping 234 b of elements, at the side of the protectionzone 112 that receives seismic power transmitted (leaked) through theprotection zone 112. The elements 102 a on the incident side, in theincident structural grouping 234 a, and corresponding elements on thetransmission side, in the transmission structural grouping 234 b, formrespective, structural arrangements similar to the arrangement 100 shownin FIG. 1B.

The element rows of the respective groupings that are closest to theprotection zone 112 may be considered to form respective structurearrangements, while the remaining, outer rows can be considered to formrespective wider structural arrangements. However, alternatively, aninner row from one grouping may be considered to correspond to an outerrow from the opposite grouping, and vice versa, such that all structuralarrangements have similar aperture sizes.

As can also be seen in FIG. 2D, the protection zone 112 encompasses, atleast partially, upper apertures of respective structural arrangements.It should be noted that, while the elements 102 a and 102 b each formregular, periodic arrays of elements, in other embodiments, the elements102 a and 102 b can each have other formations, and do not need to besituated in element rows. Furthermore, it should be understood that, inother superstructures within the scope of the present disclosure, therecan be unequal numbers of elements in an incident structural groupingand a transmission structural grouping. In particular, respectiveincident-side elements and transmission-side elements can still formupper apertures of one or more respective, structural arrangements,similar to that shown in FIG. 1B.

FIG. 2E is a top-view illustration of a superstructure 240 b thatincludes two structural arrangements. In particular, one structuralarrangement is formed by a first of the elements 102 a and a first ofthe elements 102 b, while a second structural arrangement similar tothat shown in FIG. 1B is formed by a second of each of the groupings 102a and 102 b of elements.

FIG. 2F is a top-view illustration of a superstructure 240 c thatincludes trench elements 203 a and 203 b. In particular, the top view inFIG. 2F shows that the trenches 203 a and 203 b are not boreholes, butinstead are substantially rectangular in their cross-sectional profiles.Each of the trench elements 203 a-b defines an inner volume in theearth, filled with a medium that resists transmission of the seismicwave 114 toward the protection zone 112. Furthermore, one pair of thetrench elements 203 a-b forms a structural arrangement similar to thearrangement 100 in FIG. 1B, while a second pair of the elements 203 aand 203 b forms a second structural arrangement. The trench elements 203a and 203 b form two separate structural arrangements, with respectiveupper and lower apertures, similar to the upper and lower apertures, 104and 106, respectively, in FIG. 1B, for example.

FIG. 3 is a cross-sectional side-view diagram showing how embodimentscan be advantageously used to protect structures in the sea, such as asea platform structure 336. The platform 336 stands above a surface 338of the sea, while legs of the platform are anchored into the earth 118below the seafloor 337. It is desirable to protect the sea platform 336,which can include an oil rig or other structure in the sea, by forming aprotection zone 312 around the platform.

In FIG. 3, the protection zone 312 is formed at the upper aperture ofelements 302 a and 302 b. The elements 302 a-b are anchored into theearth 118 at their respective lower ends 110, while their upper ends 108extend above the sea surface 338. In other embodiments, the lower ends110 of the elements 302 a-b are not anchored into the earth, butinstead, the elements 302 a-b are suspended mechanically, such that theyextend only a certain distance below the surface 338 of the sea. Ineither case, the elements 302 a-b redirect power from an incidentelastic, water wave that would otherwise be expected to be incident atthe sea platform 336 in its full strength.

The elements 302 a-b each form an inner volume containing therein amedium resistant to passage of the wave 314. For the case of waterwaves, this material can be a solid, for example. In this way, theelements 302 a-b may be formed of wood, metal, or another structure. Theelements 302 a-b can be similar to pylons, for example.

It should be understood that embodiment elastic wave damping structuresthat are used to protect against water waves are not limited to a singlestructural arrangement, such as the one shown in FIG. 3. In otherembodiments, any one of the groupings of arrangements or discretesub-elements that are illustrated in FIGS. 2B-2F may be used to protectagainst water waves, for example. Furthermore, in some embodiment wavedamping structures, only a plurality of elements 302 a on the incidentside of the protection zone are used, such that they form an incidentstructural grouping. Furthermore, as described hereinafter in connectionwith FIG. 4B, for example, structural groupings of elements can be usedto protect against damaging water waves expected to be incident on land,such as tsunami waves.

FIG. 4A is a cross-sectional, side-view illustration of incident andtransmission structural groupings 434 a and 434 b, respectively, ofelements on either side of a protection zone 112. The incidentstructural grouping 434 includes three elements 102 a on the incidentside of the protection zone 112. Each of the elements 102 a has theupper end 108 and the lower end 110, similar to the elements describedin connection with FIG. 1B. Each element 102 a defines a line 438 aextending from the upper end 108 to the lower end 110. This line formsan acute angle 440 a (less than 90°) with a line 436 that defines atravel direction of the wave 114 toward the protection zone.

A grouping of two or more of the elements 102 a, with the acute anglefor 440 a, can form an elastic wave damping structure having an incidentstructural grouping of elements. As described hereinabove, the acuteangle 440 a serves to redirect power from the seismic wave 114 around(below) the elements 102 a. Where a depth 119 of the elements 102 a(illustrated in FIG. 1B) is sufficiently large, and where the elementsextend sufficiently below and toward the protection zone 112, theelements can attenuate power from the wave 114 that reaches theprotection zone. Preferred dimensions and orientation of the elements102 a-b can be understood, in part, by reference to an acoustic“muffler” described hereinafter in connection with FIG. 12.

The elements 102 a also include an optional structural lining 439between the inner volume of the element and the exterior of the element(the earth 118). Structural linings may be helpful when used in certaintypes of earth where there is danger of the structural elementscollapsing, or where there is danger of the structural elements fillingwith water in an undesirable manner, for example. Such structurallinings can include PVC, other types of plastic, metal jackets, or anyother suitable type of lining material known in the art of civil andmechanical engineering, for example.

While an incident structural grouping, such as the grouping 434 a, canprovide some protection, it may be useful to include the transmissionstructural grouping of elements 434 b, which is optional. The grouping434 b includes elements 102 b, with a line 438 b extending from theupper end thereof to the lower end thereof forming an obtuse angle 440 bwith the line 436 defining the direction of travel of the elastic wave.In this way, an upper aperture and a lower aperture are formed, with theprotection zone 112 at the upper aperture. The lower aperture is smallerthan the upper aperture, thus preventing leakage of seismic power upinto the protection zone 112. It should be understood that, while theincident structural grouping 434 a is oriented with successive elementsoriented along the X direction, arrays of elements oriented along the Ydirection, such as those illustrated in FIGS. 2B and 2D-F providefurther advantages, including wider protection zones.

While a superstructure is not specifically annotated in FIG. 4A, itshould be understood that the incident grouping 434 a and transmissiongrouping 434 b together can be considered to form a superstructure 435.Such a superstructure is particularly well suited to preventing powerfrom incident waves from entering the protection zone 112. Combinationsof incident and transmission groupings of elements can significantlyreduce a depth to which borehole or other elements need to be drilled inorder to reduce seismic power by a given amount. This has significantcost advantages, as increasing bore depths are significantly expensive.In many embodiments, significant attenuation, such as 34 dB ofattenuation, is expected, even where depths of lower apertures in theearth, or below the surface of water, are only on the order of 100 m,for example. The incident structural grouping 434 a and optionaltransmission grouping 434 b together form a superstructure 435.

FIG. 4B is a cross-sectional side-view illustration of an incidentstructural grouping 437 of elements 402 that can be employed to protectagainst incident elastic water waves 314, such as a tsunami wave. Theelements 402 are anchored into the earth 118 near a shore of the seaexpected to receive an incident wave. The elements 402 can be solid,similar to the elements 302 a and 302 b illustrated in FIG. 3, forexample.

FIGS. 5-7 are top-view illustrations of additional incident structuralgroupings of elements with different configurations and functions. Itshould be understood that optional transmission-side structuralgroupings of elements can have similar configurations and may form partof other embodiment elastic wave damping structures, even thoughtransmission groupings are not illustrated in FIGS. 5-7.

FIG. 5 is a top-view illustration of an arrangement of elements 102 aforming an incident structural grouping 534 of the elements in asawtooth pattern 542. The incident structural grouping 534 is arrangedbetween the incoming, expected incident seismic wave 114 and theprotection zone 112. As described hereinafter in connection with FIG.9E, for example, substantially sawtooth-type groupings of elements canbe situated on both the incident side of the protection zone at whichthe anticipated seismic wave is expected to the incident, and also onthe transmission side of the protection zone, the side of the protectionzone opposite the incident side. Thus, with sawtooth-type grouping ofelements on both sides of the protection zone, elements from respectivegroupings can form one or more structural arrangements similar to thearrangement 100 in FIG. 1B. Therefore, superstructures includingsawtooth-type groupings, just as superstructures including element rowsor element array-type groupings, can also have the significant advantageof providing relatively broadband protection against incident seismicwaves.

FIG. 6 is a top-view illustration of a substantially sawtooth periodicpattern 642 of elements 102 a that form an incident structural grouping634 of elements.

The substantially periodic, substantially sawtooth pattern 642 has theadditional advantage of including more than a single row of elements inorder to provide additional attenuation. Furthermore, themultiple-sawtooth pattern can extend a greater width along the y-axis,for example, thus providing a wider protection zone 112.

FIG. 7 is a top-view illustration of an additional incident structuralgrouping of elements 102 a that illustrates interference of reflectedwavelets from the elements of a grouping. In particular, the seismicwave 114 impinges on the incident grouping of elements 102 a with anincident seismic wavefront 746. As is understood in the art of wavemechanics, the respective elements 102 a will reflect some of theseismic power incident on them in the form of seismic wavelets 748. Thereflected seismic wavelet 748 from the respective elements 102 a willinterfere with each other, either constructively or destructively invarious positions that depend on separation of the elements 102 a, aswell as the wavelength of the incident seismic wave.

The reflected seismic wavelength wavelets 748 also can interfereconstructively or destructively with the incident seismic wave 114,creating zones of the incident region in which seismic power ispotentially greater than that which is incident, and also regions inwhich incident and reflected waves destructively interfere with eachother to diminish significantly the intensity of seismic waves that arepresent in a given position. This effect can be exploited to protectcertain positions on the incident side of the protection zone 112 havingelements that are desired to be protected.

Interference effects can also be exploited advantageously to generateelectrical power electro-mechanically. In particular, as illustrated inFIG. 7, an electromechanical power generator 744 is located at aposition of expected constructive interference between the incident andreflected waves, such that the magnified mechanical power from the wavesis converted into electrical power in order to provide power during apower outage due to an earthquake causing the seismic wave, for example.

Finite element modeling has been employed to predict attenuation ofwaves of various embodiments seismic wave damping structures. FIGS.8A-8C, 9A-9B, and 9C-9D illustrate some of these methods and results.

FIG. 8A is a top-view illustration of a protection zone 112 formedbetween model borehole elements 802 a and 802 b. The model boreholeelements 802 a are arranged in element rows and include an incidentgrouping of elements 834 a. Similarly, the modeled elements 802 b arearranged in rows on the opposite side of the protection zone from theincident grouping to form a transmission grouping 834 b. As alsoillustrated in FIG. 8A, a 3D section of earth with a array of theborehole elements 802 a therein is an example of a “metamaterial” asused herein.

FIG. 8B is a three-dimensional (3D) illustration of what is shown inFIG. 8A. In particular, it is noted that the incident grouping 834 a ofelements 802 a, together with the transmission grouping of elements 802b, form a superstructure 840, with the protection zone 112 therebetween. Individual locations in the 3D meshing represent points usedfor the finite element analysis.

FIG. 8C is a 3D view of trench elements 803 a and 803 b in a 3D finiteelement meshing. The modeled trench element 803 a is similar to one ormore of the trench elements 203 a illustrated in FIG. 2F. Likewise, themodeled trench element 803 b is similar to one or more of the trenchelements 203 b illustrated in FIG. 2F.

The finite element analysis performed on the analytical modelsillustrated in FIGS. 8A-8C indicate that the borehole based seismic wavedamping structure of FIGS. 8A-8B, and the trench-based seismic wavedamping structure illustrated in FIG. 8C reduce direct seismic wavepower reaching the protection zone by more than 40 dB. It is noted thata diffractive component upwelling through the V-shaped structure hasbeen calculated to be 22 dB lower than the peak seismic wave powerobserved for the same location in the protection zone without theimplementation of the damping structures.

FIG. 9B is a 3D representation of the finite element analysis, with thedirection of the incident wave 114 shown.

FIG. 9C is a graph showing a source time function for rupture velocityof an earthquake measured in California in 1991. This source timefunction was used as input to the finite element analysis. The seismicevent is modeled for the response of the source function estimated forthe Hector Mine earthquake in 1991 (Magnitude 7.1-USGS). Typical seismicfrequencies are less than 1 Hz with minimal power above 1 Hz.

FIG. 9D is a graph showing power reduction calculated in a foundation912, both with and without the borehole cloak (substantially sawtoothperiodic pattern superstructure illustrated in FIG. 9E formed ofelements 903 a in a substantially sawtooth 942 a grouping and elements903 b in a substantially sawtooth 942 b grouping).

FIG. 9E is collection of areal and depth view seismic wave snapshotscalculated with and without embodiment seismic wave damping (cloaking)structures, providing a finite difference model of the effects onseismic wave propagation from seismic cloaking. In particular, the toprow shows an areal view of seismic wave snapshots, with and withoutcloaking structures. The bottom row shows a depth view of seismic wavesnapshots, with and without cloaking structures.

FIG. 9E illustrates that using a single vertical borehole array ortrench may significantly reduce the surface wave power reaching aprotected region. However, seismic power is able to be directed bydiffraction around the barrier. Using a V-shaped muffler (sawtooth)design is, therefore, much more effective in blocking surface waves inthe 3D extent.

FIG. 10 is a table showing various high-value assets that may beprotected, as examples, using embodiment elastic wave dampingstructures. FIG. 10 also shows an expected upper aperture widths (see,e.g., upper aperture 804 in FIG. 8B) and lateral extents (see, e.g.,lateral extent 805 in FIG. 8A) of a damping structure for each exampleasset, with a corresponding, example number of boreholes that isexpected to be effective in significantly attenuating seismic powerwithin a protection zone, such as the protection zone 112 in FIGS.8A-8B, including the asset.

Superstructures as described herein, and as exemplified by thesuperstructure 840 in FIG. 8B, can divert, attenuate, and createdestructive interference of hazardous seismic waves that would otherwisereach a high valued asset such as the assets listed in FIG. 10. Examplesuperstructures, also referred to herein a cloaking arrays, may be50-200 meters wide (in upper aperture), and hundreds of meters to a fewkilometers in lateral extent depending on protected asset size and risk.An example, single borehole diameter can be 1 meter for the structureslisted in FIG. 10, where 400 boreholes can populate an example 100square meter region. For most applications, borehole depths can be anestimated 150 meters or less. These depths can be particularly relevantwhere surface seismic waves are of greatest concern.

FIG. 11 is a series of diagrams 1160-1162, and a graph 1163, showing thecomparative effects of seismic cloaking generally on seismic wavepropagation for a single, vertical barrier element compared with anembodiment structural element, as calculated using a finite difference2D model of the barrier structures. The seismic event is modeled for theresponse of the source function estimated for the Hector Mine earthquakein 1991 (Mag. 7.1-USGS), as illustrated in FIG. 9C. Typical seismicfrequencies are less than 1 Hz, with minimal power above 1 Hz.

In particular, the diagram 1160 shows an areal view of seismic wavesnapshots without cloaking, while the diagram 1161 shows the effect of asingle, frontal vertical borehole 1164. Further, the diagram 1162 is adepth view snapshot of the effect of cloaking using a structuralarrangement including two angled borehole elements 102 a, 102 b asdescribed in relation to FIG. 1B.

As illustrated in the comparison, using a single vertical borehole arrayor trench may significantly reduce the direct surface wave energyreaching a protected region. However, energy is still able to diffractaround the barrier and enter the protected region. Using a V-shapedmuffler (structural arrangement) formed of elements 102 a and 102 b ismuch more effective in blocking surface waves in the 3D extent. Thegraph 1163 of seismic power as a function of time reaching theprotection zone further bears out this fact. A curve 1164 corresponds tograph 1160, with no boreholes and the highest power; a curve 1165corresponds to graph 1161 with one vertical borehole and somewhatdiminished power reaching the protection zone. However, a curve 1166corresponds to the angled boreholes case of graph 1162, with greatlydiminished power entering the protection zone between the angularboreholes.

FIGS. 12A- 12 B are a diagram and equations illustrating semi-analyticalsimulation of acoustic propagation through a “tapered muffler” 1200geometry. The tapered muffler 1200 and associated equations can be foundin Easwaran and Munjal, J. Sound and Vibration 152 (1992), 73-93 and canbe used as part of understanding the efficacy of embodiment structuralarrangements, such as the arrangement 100 illustrated in FIG. 1B, inproving a barrier against seismic waves.

The muffler 1200 includes a tapered funnel with a subwavelength opening(entrance; lower aperture) 1206 with respect to a wavelength of anacoustic radiation source 1214. The mathematical expressions shown inFIG. 12B provide a simple, analytical, transfer matrix approach toevaluating energy attenuation as a function of the entrance 1206, anexit (upper aperture) 1204, a depth 1219, and related angle dimensions.

FIGS. 13A and 13B are graphs showing transmission loss as a function offrequency for various acoustic “muffler” parameters assuming a wavespeed of 1500 meters per second (m/s). The calculations were performedusing the analytical tools shown in FIGS. 12A-12B in order to at leastbegin to understand optimization of parameters for structuralarrangements such as the arrangement 100 in FIG. 1B.

In general, the calculations for mufflers suggest that small entrance,shallow depth and steep angle can all help to maximize attenuation ofacoustic waves incident at the entrance aperture. Specific trendsobserved in calculations such as those shown in FIGS. 13A-13B include:(i) Increasing the depth at constant angle increases attenuation at thelower frequencies, but reduces attenuation at the higher frequency dueto multiple nodes; (ii) Increasing the depth at constant top & bottomaperture dimensions (i.e., while reducing the angle) reduces attenuationacross the entire frequency range; and (iii) Increasing the bottomaperture opening reduces attenuation across the entire frequency range.These trends can be applied advantageously to design of structuralapertures for seismic wave attenuation in embodiment elastic wavedamping structures in order to maximize seismic wave attenuation.

FIG. 14 is a collage of photographs showing apparatus used to study, ona model scale, non-naturally occurring, man-made structures such asborehole arrays and trenches embedded in elastic media analogous to rockand compact soil using a machined table-top scaled physical model. Theeffort focused on examining the effects of borehole arrays and trenches(metamaterials) on seismic wave propagation, diversion, scattering, andattenuation through spatial measurements from controlled seismicsources. The time series measurements were then compared and analyzedwith computer model simulations.

The solid model was composed of Delrin® plastic 1464 with a P-wave speedof 1700 m/s, S-Wave speed of 855 m/s, and a density of 1.41 g/cm3.Delrin® blocks were machined to contain boreholes in prescribed patternsor trenches defining a V-shaped muffler and compared with homogeneoussolid blocks. The Delrin® blocks contained boreholes in prescribedpatterns or trenches defining a V-shaped muffler.

The model muffler was aimed at significantly reducing the elastic wavepower reaching a ‘protected keep-out’ zone from a controlled elasticwave source. Each borehole had a diameter of 3 mm and was separated 3 mmapart from neighboring boreholes, forming a single line, where the lineextended the entire length of the block. A near and far borehole lineV-shaped pattern was formed where the near and far borehole line spacingis 3 inches apart on the Delrin® surface, the boreholes are sloped witha 5 inch length (4 inch vertical depth), and provided an apertureopening at the V-borehole barrier structure base of 0.5 inches.

Similarly, a V-trench barrier structure was machined in a separateDelrin® block where the 3 mm diameter boreholes were in contact, formingcontinuous hollow walls on both sides of the barrier structure. AModal-Shop® variable transducer 1462 was used to vertically load on theDelrin® block surface to prescribed loading functions. A 10 kHz Rickerwaveform was used to act as the seismic input function to generate theelastic wave propagation in the Delrin® blocks. PCB® model 352C33accelerometers 1460 were used to measure the temporal and spatialvibration distributions observed on the Delrin® surface. An IOTECH®wavebook 516E was used to record each time series trace using asynchronized 70 kHz sample rate per channel.

FIGS. 15A-15C show measured results that illustrate the effects of amodel V-trench machined in Delrin® block table-top experimentalconfiguration. FIG. 15A shows four accelerometer time series tracesmeasured in a center line across a homogeneous solid Delrin® blockrelative to the transducer source location. Receivers 1, 3, 4, and 7were 1, 3, 4, and 7 inches from the source, respectively. Particlevelocities were computed by integrating the measured accelerations andthen applying a high-pass filter to remove low frequency drift. A single10 kHz Ricker vertical load burst was recorded as it traveled from itssource. Each trace records a similar time series, showing a directsurface wave arrival (circled outline) followed by later reflectionarrival interference from the Delrin® block side and bottom boundaries.The first break of the direct arrivals shows a wave speed of 1693 m/s.Spherical spreading and Delrin® attenuation losses are not compensatedin the measurement plots. At the observed wave speed, the P-wavelengthwas estimated at 17 mm.

FIG. 15B shows four accelerometer time series traces measured in thecenter line across the Delrin® block that contains a V-trench barrierstructure perpendicularly oriented to the direction of elastic wavepropagation relative to the transducer source location. Receivers 1, 3,4, and 7 were 1, 3, 4, and 7 inches from the source, respectively, wherereceivers 3 and 4 were between the near and far trench walls.

FIG. 15C shows the trench barrier structure in schematic form. The timesseries traces show that the direct surface wave is reflected off thenear trench wall, where very little direct wave is observed inside thekeep out zone between the near and far trench walls. The reflectedarrival interference from the bottom surface is observed at the surfacebetween the trench walls. In this geometry, elastic waves were able toleak through the aperture at the trench bottom and travel to thesurface. These amplitudes, however, are lower than those of the peaksurface wave that would be observed in the same locations if the barriercloaking structure were not present.

FIG. 16 shows earthquake magnitude reduction expected due to subsurfacebarrier structures, based on extrapolation from the measured andmodelled results. In this simple analysis, the power drop observed inthe measurement and model studies are presented in terms of Mwreduction. The V-trench structure shows that a magnitude 7.0 earthquakeenergy intensity can be reduced to 5.4-5.0 for the peak power of thedirect destructive surface wave. The leakage through the aperture ismeasured to show a modest reduction. However, when modeling the earth,where the boundaries are infinite, diffraction leakage through theaperture is small, and the structure would be expected provide asignificant reduction in wave energy. Notably, modeled and measuredDelrin® block waveforms and amplitudes agreed within 3 dB.

Seismic Wave Damping Systems and Corresponding Methods

In addition to the seismic wave damping structures describedhereinabove, embodiments may include seismic wave damping systems thatprovide significant benefits beyond the benefits of the structuresalone. Seismic wave damping structures, together with anti-resonancedamping structures described herein, have particular synergistic effectswhen used in combination with each other.

On one hand, seismic wave damping structures described above can providesignificant protection against damaging seismic power, particularlyagainst the most damaging, lower-frequency range of seismic waves. Onthe other hand, separately, various alternative types of earthquakeprotection that may be generally less-effective against lower-frequencywaves have been proposed, including Helmholtz resonators, tree or towerarrays, meta-concrete arrays, etc. A problem of the various proposedalternative types of protection is that they may particularly effectiveonly at particular frequencies for which they are designed. Thus, thesealternatives may not provide protection for a given expected earthquake,whose specific seismic frequencies may not be known in advance. Further,while various proposed alternative types of protection may be built inthe same location to address different frequencies, engineering in thismanner is redundant and expensive.

A problem of the seismic wave damping structures described above is thatthey may be characterized by one or more resonant frequencies that maypropagate well within a protection zone defined by the structures. Thepresent inventors have recognized that this problems of both the seismicwave damping structures and the various proposed alternative types ofprotection may become special advantages in a system that incorporatesboth a seismic wave damping structure, as described hereinabove, andalso one or more of the various proposed alternative types of earthquakeprotection, which are referred to in the context of the embodimentsystems of the present application as “anti-resonance dampingstructures” and by other terms describing specific examples thereof

An advantage of the seismic wave damping structures describedhereinabove is that their resonant frequencies may be predicted inadvance. Then, with that knowledge, it is not only helpful, but alsoespecially advantageous to configure an anti-resonance damping structureto address the particular resonant frequencies supported by the seismicwave damping structures. Thus, when used together, the two structuresact synergistically to increase protection against seismic waves insignificant manner, beyond the benefits that could be expected fromeither seismic wave damping structures or anti-resonant dampingstructures acting individually.

Further, when combined as described herein, the benefits are beyondthose that may be predicted simply from a hypothetical combination ofany two given earthquake protection structures. As noted above, both theseismic wave damping structures and the anti-resonance dampingstructures have disadvantages when used separately, namely thathigher-frequency resonances are allowed to propagate, and that onlyprotection against particular frequencies is generally provided,respectively. However, the inventors have realized that when used incombination, these very disadvantages each become advantageous.Particularly, there is a way described to predict resonance frequenciesthat may be supported by a seismic wave damping structure, and insteadof attempting to engineer an alternative earthquake protection structureto protect against primary (incident) seismic waves, an alternativestructure (“anti-resonance damping structure,” as used herein) may beengineered to protect against specific resonant frequencies predictedfor the seismic wave damping structure.

Based on properties of a host medium in which the seismic dampingstructure is embedded, resonance frequencies supported by a givenseismic wave damping structure may be specifically predicted based onone or more properties of the host medium and on one or more physicalproperties of the seismic damping structure. A correspondinganti-resonance damping structure may, therefore, be configured to dampenspecifically a residual wave propagating within the protection zone atthe resonance frequency. A synergy of this arrangement is that theanti-resonance damping structure need not be configured to address allpossible seismic frequencies, potentially requiring more complex andextensive engineering. Instead, the anti-resonance damping structure maybe built and configured to dampen, specifically, only one or moreresonance frequencies supported by the seismic damping structure, wherethese resonance frequencies may be specifically predicted based onproperties of the host medium and seismic damping structure.Accordingly, according to embodiments described hereafter, particularsynergies may be obtained, which have not been recognized previously.

As noted hereinabove in the Summary section, a certain amount of seismicwave power will be able to enter a disclosed seismic damping structure.In FIG. 1A, this seismic wave power is referred to as “attenuatedsurface waves 60′,” as opposed to the incident surface waves 60 that areinitially incident at the elements 30 a-b of the seismic dampingstructure. Hereinafter, this residual seismic wave power propagatingwithin the protection zone may be referred to as “residual waves,” whichmay be residual surface waves or other types residual seismic wave powerthat enter the protection zone. The residual wave power that propagateswithin a protection zone, such as the protection zone 40 illustrated infor FIG. 1A, will propagate most strongly at seismic wave frequenciesthat are resonant in the seismic damping structure.

FIG. 17A is similar to FIG. 1A in some respects, but also illustrates aseismic wave damping system, in its environment of use, the benefits ofwhich extend beyond those described above for seismic damping structuresalone. FIG. 17A also includes terminology specific to seismic dampingsystems disclosed herein, as further described hereinafter. In FIG. 17A,in addition to the elements 30 a and 30 b, which constitute a seismicdamping structure, there is illustrated an anti-resonance dampingstructure 1770 that dampens residual wave 1760′ that propagate withinthe protection zone at resonance frequencies at a resonance frequencydefined by the seismic damping structure. As seismic waves 1760 travelfrom the earthquake center 50 in a distal medium 1719, the seismic wave1760 are incident at the element 30 a, which forms a border of theprotection zone 40 together with the element 30 b. The anti-resonancedamping structure 1770 is positioned within the protection zone 40 andis configured to dampen the residual wave 1760 from 1760′ propagatingwithin the protection zone at the resonance frequency. Example resonancefrequencies are illustrated hereinafter in connection with FIG. 24E, forexample.

Example anti-resonance damping structures 1770 may include a Helmholtzresonator or array of Helmholtz resonators, as illustrated in FIG. 23Aand FIG. 23D, an array of trees planted in the protection zone, or anarray of towers placed on the protection zone, as illustrated in FIGS.24A and 24D, and array of buried cylinders, such as the meta-concretearray illustrated in FIGS. 25A and 25C, among other structures that aredescribed herein or are otherwise or are known to those of skill in theart.

FIG. 17B is a schematic diagram illustrating the elements 30 a and 30 b,which form a seismic damping structure 1700, together with theanti-resonance damping structure 1770, as illustrated in FIG. 17A.Together, the structure 1770 and the seismic damping structure 1700 forman embodiment seismic wave damping system 1772. It should be understoodthat seismic damping structures (also referred to herein by “seismicwave damping structures,” “elastic wave damping structures,”“superstructures,” “arrangements” of elements, and similar terms) thatare described throughout the application may be used in an embodimentseismic wave damping systems. These structures include structuralarrangements, such as the arrangement 100 in FIG. 1B, superstructures,such as the superstructure 240 a illustrated in FIG. 2D, incident andtransmission groupings of elements as described in connection with FIGS.6 and 8A-8C, for example, and any other structures described herein. Foreach of these structures, in view of the disclosure herein, a person ofskill will understand how to calculate resonance frequencies from aproperty of the host medium and from one or more properties of theelements embedded within the host medium, such as depth or lengththereof. A depth 119 of an element 102 b, for example, is illustrated inFIG. 1B. A further property of an element may include a length thereof,such as a length from the upper and 108 to the lower and 110 of theelement 102 a illustrated in FIG. 1B.

The host medium 1718 shown in FIG. 17A can include earth, such as theearth 118 illustrated in FIG. 4A. Furthermore, any natural or man-madehost medium, such as rock, sand, gravel, etc. or concrete, bricks, orother building materials may form part of the host medium 1718.Furthermore, it should be understood that the distal medium 1719 may beof the seven of the same material as the host medium 1718, forming onecontinuous medium. The distal medium and host medium are delineated onlyby the seismic damping structure, comprising the elements 30 a and 30 bin FIGS. 17A-17B, which form the boundary of the protection zone.

It should be further understood that the anti-resonance dampingstructure 1770 may be configured to dampen the residual wave 1760′ atthe resonance frequency by being tuned to dampen, and being mechanicallytuned to a harmonic or subharmonic of the resonance frequency.Furthermore, in some embodiments, anti-resonance damping structures maybe configured to dampen a residual wave by being mechanically tuned tothe resonance frequency, one or more harmonics of the resonancefrequency, one or more sub harmonics of the resonance frequency, or anycombination thereof. An example harmonic of a resonance frequency of 5Hz is 10 Hz, for example. An example subharmonic of a resonancefrequency 5 Hz is a frequency of 2.5 Hz, for example.

FIG. 18A illustrates general “seismic muffler” geometry for disclosedseismic damping structures described hereinabove. In the example of FIG.18A, a borehole array muffler, also referred to herein as a seismicdamping structure 1800, is configured to dampen seismic waves and form aprotection zone around a structure asset 20, in this case a nuclearpower plant, to be protected. Also illustrated in FIG. 18A is theanti-resonance damping structure 1770 of FIG. 17A, which can furtherenhance seismic protection in the system formed by the combination ofthe structure 1800 and the anti-resonance damping structure 1770 bydamping resonance frequencies defined by the seismic damping structure1800. The structure 1800 is a V-shaped seismic muffler that diverse,dissipates, and reduces ground motion from hazardous seismic waves priorto reaching the critical infrastructure 20. Seismic wave types caused byearthquakes can include surface and body waves. Surface seismic waves,such as Rayleigh and Love waves, commonly cause significant damage anddestruction to buildings and structures.

These resonance frequencies characterizing an example V-shaped muffler(seismic wave damping structure) may be determined by numericalmodelling, as will be understood by one of skill in the art in view ofthis disclosure. Alternatively, the resonance frequencies may becalculated more easily as follows. We separate the description of thetotal transmission loss (Eqn. (1)) into the entrance loss TL_(ent)(Kinsler et al., 2000) and the loss through the conical muffler TL_(con)(Easwaran and Munjal, 1992). In terms of seismology, the entrance lossis the frequency dependent reflected wave power that does not coupleinto the muffler inlet. The transmission loss through the muffler as thecoupled wave travels within the muffler to its outlet (exit) isattenuation.

TL=TL _(con) +TL _(ent)   (1)

The entrance loss is caused by the acoustic impedance of the open pipewith the subwavelength opening, as described by the lumped elementtheory (Kinsler et al., 2000).

$\begin{matrix}{{T\; L_{ent}} = {{- 10}\; {\log \left( {1 - {\frac{{0.25\left( {k_{0}b} \right)^{2}} + {0.6{i\left( {k_{0}b} \right)}} - 1}{{0.25\left( {k_{0}b} \right)^{2}} + {0.6{i\left( {k_{0}b} \right)}} + 1}}^{2}} \right)}}} & (2)\end{matrix}$

Furthermore, the transmission loss through the conical muffler can beexpressed in terms of a transfer matrix relating the pressure andvelocity at muffler inlet (entrance) (p1, v1) to those at the muffleroutlet (exit) (p2, v2).

$\begin{matrix}{\begin{bmatrix}p_{1} \\v_{1}\end{bmatrix} = {{\begin{bmatrix}T_{11} & T_{12} \\T_{21} & T_{22}\end{bmatrix}\begin{bmatrix}p_{2} \\v_{2}\end{bmatrix}}.}} & (3)\end{matrix}$

Using the above transfer matrix, we can describe the TL_(con) term inEqn. (1) for a conical muffler as follows (Easwaran and Munjal, 1992):

$\begin{matrix}{{{T\; L_{con}} = {20\; {\log\left( {0.5\frac{b}{t} \times {{T_{11} + \frac{4{vT}_{12}}{\pi \; t^{2}} + {T_{21}\frac{4v}{\pi \; b^{2}}} + {\left( \frac{t}{b} \right)^{2}T_{22}}}}} \right)}}},} & (4)\end{matrix}$

in which the terms in the Eqn. (4) are in turn defined below:

$\begin{matrix}{{T_{11} = {{\frac{z_{2}}{z_{1}}{\cos \left( {k_{0}l} \right)}} - \frac{\sin \left( {k_{0}l} \right)}{k_{0}z_{1}}}},} & (5) \\{{T_{12} = {i\frac{4{vz}_{1}}{\pi \; b^{2}\; z_{2}}{\sin \left( {k_{0}l} \right)}}},} & (6) \\{{T_{21} = {{i\frac{\pi \; b^{2}{\sin \left( {k_{0}l} \right)}}{4v}\left( {\frac{z_{2}}{z_{1}} + \frac{1}{\left( {k_{0}z_{1}} \right)^{2}}} \right)} - {i\frac{\pi \; b^{2}{\cos \left( {k_{0}l} \right)}}{4{vk}_{0}z_{1}^{2}}}}},} & (7) \\{{T_{22} = {{\frac{z_{1}}{z_{2}}{\cos \left( {k_{0}l} \right)}} + \frac{\sin \left( {k_{0}l} \right)}{k_{0}z_{2}}}},} & (8) \\{{z_{1} = \frac{bl}{t - b}},} & (9) \\{{z_{2} = {z_{1} + l}},} & (10) \\{k_{0} = {\frac{2\pi \; f}{v}.}} & (11)\end{matrix}$

Term definitions for the muffler characteristics specified in Eqns.(1)-(11) are described in Table 1.

TABLE 1 Term Definitions for Muffler Characteristics in Equations (1) to(11) Equation Term Definition TL Total transmission loss through mufflerstructure TL_(ent) Transmission loss across muffler inlet TL_(con)Transmission loss through the conical muffler P_(i(1, 2)) Pressure atmuffler inlet (entrance) and outlet (exit) v_(i(1, 2)) Elastic wavevelocity at the muffler inlet (entrance) and outlet (exit) k₀ Wavevectorin free space, defined in equation (11) B Muffler inlet (entrance)diameter T₁₁, T₁₂, T₂₁, T₂₂ Transfer matrix elements, defined inequations (5-8) T Muffler outlet (exit) diameter L Muffler lengthz_(i(1, 2)) Geometrical muffler quantities, defined in equations (9-10)v Elastic wavespeed in medium F Elastic wave frequency

The above formalism of the conical muffler depicted in Eqns. (1)-(11)was used to estimate elastic wave transmission and attenuationperformance for various muffler geometries and sizes scaled to nominalareas for example infrastructure protection vulnerable to damage fromearthquakes, treating seismic waves as example elastic waves. Muffler(seismic wave damping structure) performance results calculated based onEqns. (1)-(11) are depicted in FIGS. 19A-21C. From the performanceresults, resonance frequencies characterizing seismic wave dampingstructures may be determined, as will be illustrated in examplesdescribed hereinafter.

In particular, the one or more resonance frequencies of a seismic wavedamping structure may be determined as a function of depth of theelements forming the muffler (L, as used in Table 1 and illustrated inFIG. 19B). The one or more resonance frequencies may further bedetermined as a function a physical property of the host medium such aswavespeed in the host medium, which is v in Table. 1. A particularmanner in which the equations and parameters noted above lead to adetermination of resonance frequencies will become further apparent toone of skill in the art in view of the examples provided hereinafter.

In summary, in order to achieve the maximum attenuation of seismicmotion for a broadband frequency response, the muffler parameters shouldinclude small inlet diameters, shallow depths with shorter lengths, andsteep muffler wall angle (shallow slope). In these calculations, a 25-30dB transmission loss through the muffler was possible for a shallow,steep angle structure at 1 Hz of seismic frequency, with greaterreductions at frequencies less than 1 Hz. Seismic frequencies from 0.1to 1 Hz are common for many large magnitude earthquakes. This frequencyband appears to be well suited for the muffler structure. The predictedtransmission loss at these frequencies indicates that the impact oflarger earthquakes can be reduced to a less destructive level by theemployment of conical mufflers.

Additional details regarding these calculations may be obtained in“Seismic Muffler Protection of Critical Infrastructure fromEarthquakes,” Robert W. Haupt, et al., Bulletin of the SeismologicalSociety of America, Vol. 108, No. 6, pp. 3625-3644, Dec. 2018, which ishereby incorporated herein by reference in its entirety.

FIG. 18B is a color-coded or shaded, side-view graphical drawing thatillustrates particle velocity, in unit length per second, over across-sectional area showing the seismic damping structure 1800 in crosssection.

FIGS. 19A-19D are cross-sectional diagrams and graphs, respectively,illustrating the effect of a muffler inlet diameter and wall slope onseismic wave transmission loss behavior through conical shaped mufflers(seismic damping structures for different wall slope inlet diameters.FIG. 19A specifically is a cross-sectional diagram of a seismic dampingstructure illustrating muffler geometry and parameters situated in ahomogenous, isotropic, solid host medium. FIG. 19A illustrates the samebasic geometry shown in FIG. 12A, wherein the terms entrance (loweraperture), exit (upper aperature), and length, among other terms, aredefined.

FIG. 19B is a diagram illustrating four different muffler geometries andtheir respective dimensions. These structural geometries are contrastedwhere the outlet diameter is 0.5 km and the muffler length (alsoreferred to herein as muffler “depth” or as a “depth” of elementsforming a seismic wave damping structure is 0.25 km for all cases a-d.The inlet diameter is varied from 0.1 km, to 0.25 km, to 0.35 km, to 0.5km.

FIG. 19C is a graph illustrating elastic wave transmission loss as afunction of seismic wavelength corresponding to seismic frequenciesspanning from 0.1-10 Hz. These calculations treat seismic waves aselastic, which is a helpful approximation. This is shown for all fourcases a-d corresponding to FIG. 19B.

FIG. 19D is a graph illustrating P and S wave transmission loss(spectral ratio of muffler outlet and inlet) behavior for the fourmuffler examples of FIG. 19B as a function of seismic frequency. Thesolid lines in FIG. 19B represent P-wave transmission losses with a Pwave velocity of 1550 m/s, while the dotted lines represent S wavelosses with an S wave velocity of 17th of 700 m/s.

FIGS. 20A-20C illustrate the effect of muffler length and outputdiameter for various damping structure seismic damping structuregeometries. In particular, seismic wave transmission losses throughproportional conical mufflers for different muffler length and outputand outlet diameters are illustrated, where wall slope and inletdiameter are constant. FIG. 20A illustrates four different mufflergeometries that are contrasted, where the inlet diameter is 0.1 km andthe muffler wall slope is constant for all cases. The outlet diametersare 0.1 km, 0.2 km, 0.4 km, and 0.6 km, where the corresponding mufflerlength are 0.18 km, 0.55 km, and 0.9 km, respectively. A vertical wallmuffler is shown for comparison purposes.

FIG. 20B is a graph showing elastic wave (approximate for a seismicwave) transmission loss as a function of seismic wavelengthcorresponding to seismic frequencies spanning 0.1-10 Hz.

FIG. 20C is a graph illustrating P and S wave transmission loss(spectral ratio of muffler outlet and inlet) behavior for the fourmuffler examples illustrated in FIG. 20A as a function of seismic waveincident frequency. Solid lines represent P-wave transmission losseswith a P velocity of 1500 m/s, while dotted lines represent the S wavelosses with an S velocity of 700 m/s.

FIGS. 21A-21C illustrate a comparison of the analytical calculations to2D numerical finite difference models for two muffler examples, namely ashallow sloping muffler model (a) having a wall slope and a verticalwall muffler model (d) (also shown in FIG. 19B). FIG. 21A illustratescross-sectional muffler geometry for these examples. FIG. 21B is across-sectional illustration of the example structures (a) and (d). FIG.21C shows calculated P and S wave transmission loss (spectral ratio ofmuffler outlet and inlet) behavior for the two muffler examples of FIG.21A as a function of seismic frequency. Solid lines represent P-wavetransmission losses with a P-velocity of 1500 m/s, while dotted linesrepresent S-wave losses with an S-velocity of 700 m/s. The symbolmarkings correspond to the numerical model computations for singlefrequency—continuous wave (CW) seismic tones.

FIG. 22A illustrates a source time function from the Hector Mine 1999earthquake (Mag. 7.1-USGS), which was estimated from the slip velocitydistribution. The 2D vertical-depth view and 2D aerial-plan viewsimulations described hereinafter use a source time functionproportional to the normalized slip velocity illustrated in FIG. 22A.FIG. 22B is a graph showing a frequency distribution for the Hector Mineearthquake. The source time function exhibits its peak amplitude at 0.3Hz with minimal amplitudes above 3 Hz.

FIG. 23A is a color-coded or shaded vertical depth view of the seismicwave field after nine seconds for the up-going shear waves, where aprotection zone is formed by elements 30 a-30 b, combined with aHelmholtz resonator array 2370, which together form a seismic wavedamping system 2372.

FIG. 23B is a graph showing the source function for the Hector Mineearthquake, which was injected by simulation at the bottom boundaryshown in the graph, over time, into the horizontal component as a sourceas a line source.

FIG. 23C is an equation showing a transmission factor T for a finitenumber N of resonators. In this manner, a net effect on residual wavedamping for a Helmholtz resonator array may be calculated.

FIG. 23D is a schematic diagram illustrating the Helmholtz resonatorarray, anti-resonance damping structure 2370 that is pictoriallyillustrated in FIG. 23A.

The following are two examples for how to use the equations in FIG. 23Cin order to configure a Helmholtz resonator array to address aparticular resonance frequency of a seismic damping structure. When theHelmholtz resonator array, or another anti-resonance damping structurewithin the scope of embodiments, is built or otherwise configured toaddress a particular seismic wave frequency, as used herein, it isconsidered to be configured to dampen residual waves having thefrequency by being “mechanically tuned” to the resonance frequency.“Mechanical tuning” in some embodiments may be accomplished by virtue ofconstruction with particular mechanical dimensions, properties, orparameters. In other embodiments, “mechanical tuning” may beaccomplished in whole or in part by adjustments that follow actualconstruction of the anti-resonance damping structure.

In some embodiments, anti-resonance damping of residual waves in aprotection zone may be accomplished by mechanically tuning ananti-resonance damping structure to a harmonic or subharmonic of theresonance frequency. Furthermore, as will be understood in view of thisdescription, a seismic wave damping structure may be characterized bytwo or more resonance frequencies. Accordingly, two or more resonancefrequencies for residual seismic waves may be targeted for attenuationby mechanically tuning one or more anti-resonance damping structures totwo or more respective frequencies, whether resonance frequencies,harmonics or subharmonics thereof, or any combination of these.

In the case of a seismic damping structure having a resonance frequencyat 5 Hz, the Helmholtz resonator array anti-resonance damping structuremay be built with cylinder width of 1.7 m, cylinder height of 1.3 m,neck width of 0.1 m, and neck height of 0.2 m. As a second example, fora resonance frequency of a seismic damping structure at 10 Hz, acylinder width for the Helmholtz resonator array may be 1.7 m, withcylinder height 1.3 m, neck width of 0.24 m, and neck height of 0.2 m.Additional details regarding parameters for Helmholtz resonators may befound, for example, in Wang et al., J. Appl. Phys. 103 (2008) 064907,which is hereby incorporated herein by reference in its entirety.

FIG. 23E is a graph illustrating the benefits of using the Helmholtzresonator array (anti-resonance damping structure) 2370 in the seismicwave damping system 2372. A curve 2373 shows seismic amplitude as afunction of frequency where there is no seismic wave damping structureprovided to create a protection zone. Correspondingly, where only aseismic wave damping structure is used without an anti-resonance dampingstructure, a curve 2376 shows a greatly reduced seismic amplitude as afunction of frequency. However, as will be noted, the curve 2376 showsseveral resonances 2374 (also referred to herein as “resonancefrequencies” or “resonant frequencies”), that are present in thespectrum 2376. By contrast with the curve 2376 and the curve 2373, adashed curve 2378 shows the effect of a full seismic wave damping systemthe full seismic wave damping system 2372 including the Helmholtzresonator array 2370. As illustrated, the resonances 2374 aredramatically damped at the higher frequencies associated with theseismic wave damping structure alone. This illustrates the significantdamping benefit that can be obtained by embodiments herein when a systemincludes an anti-resonance damping structure that is configuredspecifically to dampen the one or more resonance frequencies of theseismic wave damping structure.

FIG. 24A is a color-coded or shaded, vertical depth view graphs of aseismic wave field after nine seconds for the up going shear wave, wherethe source function for the Hector Mine earthquake, as illustrated inFIG. 24B, was injected at the bottom boundary over time into thehorizontal component as a line source. A representative above groundtower or tree anti-resonance damping structure 2470 is illustrated,which, together with the seismic wave damping structure formed by thecombination of elements 30 a and 30 b, forms a seismic wave dampingsystem 2472.

FIG. 24C illustrates an equation that can be used to calculate afrequency that can be damped by the array 2470 of resonators (in FIG.24A, towers or trees, for example) illustrated in FIG. 24A.

FIG. 24D is a more detailed illustration of the array of resonators2470, which have a lattice constant L and height H above the ground. Lmay have various values in different embodiments, depending on thenumber of resonators desired, the resonant frequency to be addressed,the diameter or other cross-sectional dimension of the elements, andother factors. At the right of FIG. 24D is shown a cross-sectional viewof a single one of the resonators 2469, which can be a tower or tree,for example. The resonator 2469 has a cross-sectional area A and adiameter D. The diameter D, for a cylindrical tower or tree, is anexample of a cross-sectional dimension. In general, embodiments mayinclude cross-sectional dimensions, such as diameters, on the order or 1m, on the order of 5 m, on the order of 10 m, for example. Using theinformation of FIGS. 24C-24D, a person of skill can readily calculate aresonance frequency that may be addressed by a given array of towertowers or trees, and a person of skill may configure the tower or treearray 2470 to dampen a residual wave propagating within a protectionzone at one or more given resonance frequencies. It will be understoodthat while the equation in FIG. 24C is for the infinite array ofresonators 2470, similar results may be obtained for a large array, orcalculated, or numerically obtained via numerical modeling to addressparticular resonance frequencies of associated seismic dampingstructures. An array of trees may be an array that does not naturallyoccur, such as an array of trees that are periodically spaced, forexample.

Heights H of various embodiments, extending vertically from the surfaceof the host medium, may be between a few meters and hundreds of meters,such as 100 m or less, for example. A particular advantage of embodimentsystems is that an anti-resonance damping structure need not be built toattempt to counteract the direct influence of incident seismic waves,the most destructive frequencies of which are usually at the lower endof the seismic frequency range, such as between 0.1 Hz and 3 Hz, as anexample. Instead, in embodiment systems, anti-resonance dampingstructures are configured to dampen seismic waves propagating at theresonance frequency or frequencies of the respective seismic dampingstructures. The seismic damping structures are particularly effective atdamping lower seismic frequencies, while the resonance frequencies tendto be higher, such as above 3 Hz, above 5 Hz, or above 10 Hz, forexample. For damping at higher frequencies, towers, trees, or otherstructures may generally be smaller, shorter, and more feasible andinexpensive to construct.

For exemplary wood towers, example parameters can include v_(p)=2200m/s, p_(r)=450 kg/m3, vs=1200 m/s, and A=0.071 m². Example ground hostmedium parameters can include v_(p)=900 m/s, p_(g)=1200 kg/m³, andv_(s)=500 m/s, with r=v_(s)/v_(p), L=50, μ_(g)=1×10⁸ for a shear modulusof soil, and E_(r)=1×10⁹ for Young's modulus of the wood. With theseparameters, in particular examples, where an example height of theresonators is 50 m, the resonant frequency addressed will be 5 Hz, whilein a second example, with a height of 25 m, the resonant frequencyaddressed will be 10 Hz. Furthermore, assuming that a resonant frequencyis 5 Hz for example, a height of 25 m may still be used, thus tuning thearray of resonators 2472 to dampen the harmonic frequency 10 Hz, whichwill still dampen a residual wave propagating within the protection zoneat 5 Hz. Further information regarding these calculations may be found,for example, in Colombi et al. (2016), “A seismic meta-material: Theresonance meta-wedge,” Sci. Rep. 6, 27717, which is hereby incorporatedherein by reference in its entirety.

FIG. 24E is a graph showing seismic amplitude as a function of frequencyin connection with the infinite array of resonators 2470 illustrated inFIGS. 24A and 24D. The curve 2373 shows a case where no seismic waveprotection is provided. The curve 2376, as in FIG. 23E, illustrates acase where only the seismic wave damping structure formed by theelements 30 a and 30 b is provided, wherein the structure ischaracterized by resonances the resonances 2374. However, where themuffler (seismic wave damping structure) is combined with the array oftower resonance absorbers 2470, the resonances 2374 are significantlydamped.

FIG. 25A is a color-coded or shaded, vertical depth view graph showingseismic power in the form of particle velocity as a function of depthand cross range for a seismic wave damping system 2572. In this case,similarly, the Hector Mine earthquake source function, as illustrated inFIG. 25B, was injected at the bottom border over time into thehorizontal component as a line source the system 2572 includes theseismic wave damping structure formed by the elements 30 a-30 b and alsoa meta-concrete array 2582. The meta-concrete array 2582 is an exampleof buried cylinders, and geometry of the buried cylinders, particularlybecause the meta-concrete array 2582, is illustrated further in FIG.25C.

FIG. 25C is a cross-sectional view of the meta-concrete array 2582 thatis illustrated in FIG. 25A. A particular cylinder 2580 of the array 2582is also shown in greater detail.

FIG. 25D illustrates how resonance frequency may be obtained in terms ofany lost an elastic modulus E_(S) of a soft coating of the cylinders2580 and in terms of a core size R1 of a heavy core in the cylindricalelements 2580. This equation permits the person of ordinary skill in theart to configure an array of buried cylinders to dampen resonancefrequencies predicted for a particular seismic wave damping structure.In addition, further information regarding these example structures,which may be termed “meta-concrete arrays,” may be found in“Meta-concrete: designed aggregates to enhance dynamic performance,”Journal of the Mechanics and Physics of Solids 65 (2014) 69-81, which ishereby incorporated herein by reference in its entirety.

FIG. 25E is a graph illustrating seismic amplitude as a function offrequency that can be reduced using the meta-concrete array 2582 as partof the seismic wave damping system 2572 illustrated in FIG. 25A. Thecurves 2373 and 2376 that were previously described are shown, togetherwith the route of the resonance the resonance frequencies 2374. A curve2578 (dashed) illustrates that in the case of the meta-concrete arrayanti-resonance damping structure, similar benefits may be obtained byspecifically addressing the resonance frequencies 2374 of the seismicwave damping structure by appropriate configuration of theanti-resonance damping structure meta-concrete array 2582.

FIG. 26 is a flow diagram illustrating a procedure 2600 seismic wave forconstructing a seismic wave protection zone, according to an embodimentherein. At 2684, elements are embedded within a host medium, thusdefining a seismic wave damping structure that is characterized by aresonance frequency, and also thereby forming a border of a protectionzone. The seismic wave damping structure is particularly configured toattenuate power of a seismic wave traveling from a distal medium to thehost medium and which is incident at the protection zone formed by theelements.

At 2686, an anti-resonance damping structure is positioned within theprotection zone and is configured to dampen a residual wave propagatingwithin the protection zone at the resonance frequency. As describedhereinabove, configuring the anti-resonance damping structure to dampenthe residual wave propagating within the protection zone at theresonance frequency may include determining the resonance frequency as afunction of a depth of the elements in the host medium and of a physicalproperty of the host medium.

It should be understood that the procedure 2600, in other embodiments,may be modified to use or take advantage of any embodiment structure orsystem described hereinabove.

In a further embodiment not directly illustrated in the drawings, butwhich will be clearly understood by those skilled in the art in view ofthe other illustrations in the drawings and description herein, aprocedure for seismic wave damping includes converting an incidentseismic wave propagating in a distal medium outside a protection zoneinto a residual seismic wave propagating within the protection zone atone or more resonant frequencies. The procedure further includesdampening the residual wave within the protection zone viaanti-resonance damping. The method may optionally include use orincorporation of any of the methods; elements; seismic wave dampingstructures, superstructures, or arrangements; and anti-resonance dampingstructures summarized hereinabove pertaining to other embodiments orfurther described hereinafter in relation to other embodiments. Forexample, the distal medium may be earth, and also the protection zonemay earth or another building foundation or medium. The conversion mayoccur via the incident seismic waves of any of the types describedhereinabove being incident at a seismic wave damping structure,superstructure, arrangement, or grouping formed by elements such asborehole elements embedded in the earth. A casing (i.e., liner) may beinserted into each borehole to maintain its shape and structure.Dampening the residual wave within the protection zone viaanti-resonance damping may include use of any of the anti-resonancedamping within the scope of the claims listed hereinafter or within thescope of the embodiments otherwise described hereinabove.

The teachings of all patents, published applications and referencescited herein are incorporated by reference in their entirety.

While this invention has been particularly shown and described withreferences to example embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the scope of the inventionencompassed by the appended claims.

1. A seismic wave damping system comprising: elements, embedded within ahost medium, defining a seismic damping structure, the elements arrangedto form a border of a protection zone, the seismic damping structureconfigured to attenuate power of a seismic wave, traveling from a distalmedium to the host medium, that is incident at the protection zone, theseismic damping structure characterized by a resonance frequency; and ananti-resonance damping structure positioned within the protection zoneand configured to dampen a residual wave propagating within theprotection zone at the resonance frequency.
 2. The seismic wave dampingsystem of claim 1, wherein the resonance frequency is a function of adepth of the elements in the host medium and of a physical property ofthe host medium.
 3. The seismic wave damping system of claim 1, whereinthe anti-resonance damping structure is configured to dampen theresidual wave by being mechanically tuned to the resonance frequency. 4.The seismic wave damping system of claim 1, wherein the anti-resonancedamping structure is configured to dampen the residual wave by beingmechanically tuned to a harmonic of the resonance frequency.
 5. Theseismic wave damping system of claim 1, wherein the anti-resonancedamping structure is configured to dampen the residual wave by beingmechanically tuned to a subharmonic of the resonance frequency.
 6. Theseismic wave damping system of claim 1, wherein the anti-resonancedamping structure includes two or more anti-resonance damping structuresconfigured to dampen the residual wave by being mechanically tuned totwo or more respective frequencies selected from the group consisting of(i) the resonance frequency, (ii) harmonics of the resonance frequency,and (iii) subharmonics of the resonance frequency.
 7. The seismic wavedamping system of claim 1, wherein the anti-resonance damping structureincludes one or more Helmholtz resonators positioned on or within thehost medium within the protection zone.
 8. The seismic wave dampingsystem of claim 1, wherein the anti-resonance damping structure is anarray of cylinders within the host medium within the protection zone. 9.The seismic wave damping system of claim 1, wherein the anti-resonancedamping structure is a seismic wave absorbing structure configured todampen the residual wave by absorption.
 10. The seismic wave dampingsystem of claim 9, wherein the seismic wave absorbing structure is amass-in-mass lattice.
 11. The seismic wave damping system of claim 1,wherein the host medium is earth, and wherein the anti-resonance dampingstructure includes an array of trees planted in the earth within theprotection zone and spaced periodically.
 12. The seismic wave dampingsystem of claim 1, wherein the anti-resonance damping structure includesan array of scattering components positioned periodically.
 13. Theseismic wave damping system of claim 1, wherein the anti-resonancedamping structure includes one or more towers positioned on the hostmedium within the protection zone.
 14. The seismic wave damping systemof claim 13, wherein the one or more towers are one or more flexible,steel-girded towers.
 15. The seismic wave damping system of claim 13,wherein the one or more towers have heights, extending vertically from asurface of the host medium, between a few meters and hundreds of meters.16. The seismic wave damping system of claim 13, wherein the one or moretowers have heights, extending vertically from a surface of the hostmedium, of 100 m or less.
 17. The seismic wave damping system of claim13, wherein each of the one or more towers has a cross-sectionaldimension on the order of 1 m or on the order of 10 m.
 18. A method ofconstructing a seismic wave protection zone, the method comprising:embedding elements within a host medium, thus defining a seismic wavedamping structure characterized by a resonance frequency and forming aborder of a protection zone, wherein the seismic wave damping structureis configured to attenuate power of a seismic wave traveling from adistal medium to the host medium and incident at the protection zone;and positioning an anti-resonance damping structure within theprotection zone and configuring the anti-resonance damping structure todampen a residual wave propagating within the protection zone at theresonance frequency.
 19. The method of claim 18, wherein configuring theanti-resonance damping structure to dampen the residual wave propagatingwithin the protection zone at the resonance frequency includesconfiguring the anti-resonance damping structure based on one or moreproperties of the host medium and one or more properties of theelements.
 20. A method of seismic wave damping, the method comprising:converting an incident seismic wave propagating in a distal mediumoutside a protection zone into a residual seismic wave propagatingwithin the protection zone at one or more resonant frequencies; anddampening the residual wave within the protection zone viaanti-resonance damping.